{"paper":{"title":"Positive solutions for large random linear systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jamal Najim (LIGM), Pierre Bizeul (ENS Paris Saclay)","submitted_at":"2019-04-09T09:24:48Z","abstract_excerpt":"Consider a large linear system where $A_n$ is a $n\\times n$ matrix with independent real standard Gaussian entries, $\\boldsymbol{1}_n$ is a $n\\times 1$ vector of ones and with unknown the $n\\times 1$ vector $\\boldsymbol{x}_n$ satisfying$$\\boldsymbol{x}_n = \\boldsymbol{1}_n +\\frac 1{\\alpha_n\\sqrt{n}} A_n \\boldsymbol{x}_n\\, .$$We investigate the (componentwise) positivity of the solution $\\boldsymbol{x}_n$ depending on the scaling factor $\\alpha_n$ as the dimension $n$ goes to $\\infty$. We prove that there is a sharp phase transition at the threshold $\\alpha^*_n =\\sqrt{2\\log n}$: below the thres"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.04559","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}